Two-slit experiment extended YouTube

Measurement

We now add a new component to the two-slit experiment from the last lecture, which detects whether an electron that has turned up at one of $N$ screen detectors, went through slit $A$ or $B$.

Figure 7.1 - Detecting which slit the particle went through

Figure 7.1 - Detecting which slit the particle went through

In classical mechanics we can always measure something without affecting it in a large enough way to change the result of the experiment. We can always construct some detector that doesn't have a significant influence on the system. In quantum mechanics, this isn't the case.

We could, say, place an electron near to slit $A$, which had been prepared with spin down (see figure 7.1) and when one of the particles fired from the source at $O$ passes through slit $A$ it flips the spin to up (somehow).

This means that we have to consider two-bit states, which can describe both the state of the (fired) particle and the state of the spin detector.

Entanglement

We now show that measurement at the quantum level introduces entanglement of the states describing the system.

The initial state of the system (I) is where the (fired) particle is at the origin and the spin the (measurement) electron is down.

If only the slit at $A$ is open, then any particle that arrives at one of the detectors must have come through $A$, flipping the spin of the electron to up. Thus, $$\ket{Od} \rightarrow \ket{Au} \rightarrow \ket{\psi} = \sum_{n=1}^{N}\psi_n \ket{nu}$$

If only the slit at $B$ is open, then any particle that arrives at one of the detectors must have come through $B$, in which case the spin of the electron remains down. Thus, $$\ket{Od} \rightarrow \ket{Bd} \rightarrow \ket{\phi} = \sum_{n=1}^{N}\phi_n \ket{nd}$$

If both slits are open, then we must consider the superposition, $$\ket{Od} \rightarrow \ket{Au} + \ket{Bd} \rightarrow \ket{\psi} + \ket{\phi} = \sum_{n=1}^{N}\left(\psi_n \ket{nu} + \phi_n \ket{nd}\right)$$

Straightaway we can see a difference to the previous case, where no measurement was being made. In this case we can't even consider classically adding the amplitudes because the linear superpositions are combinations of different sets of basis vectors.

Also, there are now two states in the eigenspace associated with the projection operator of the system, which describes both the measurement of which the particle goes through and the $m^\text{th}$ detector that the particle arrives at , $$\Pi = \ket{mu}\bra{mu} + \ket{md}\bra{md}$$

Thus, the probability that a particle arrives at the (or average number of particles) is $$\begin{align*}\left(\bra{\psi} + \bra{\phi}\right)\Pi\left(\ket{\psi} + \ket{\phi}\right)&= \sum_{k=1}^{N}\left(\bra{ku}\psi_k^* + \bra{kd}\phi_k^*\right)\left(\ket{mu}\bra{mu} + \ket{md}\bra{md}\right)\sum_{n=1}^{N}\left(\psi_n \ket{nu} + \phi_n \ket{nd}\right)\\&= \sum_{k=1}^{N}\left(\bra{ku}\psi_k^* + \bra{kd}\phi_k^*\right)\left(\psi_m\ket{mu} + \phi_m\ket{md}\right)\\&= \psi_m^*\psi_m + \phi_m^*\phi_m\end{align*}$$

This is the same as the classical result - with no interference term - which we got only when we didn't try to measure which slit the particle went through. There are two ways to look at this,

  • making a measurement collapses the wave function, destroying the interference pattern
  • measurement introduces entanglement into the system, destroying the interference pattern

There are many variations we could consider. For example,

  • if the detector only had a probability of being flipped by the passing particle, then the interference pattern would only be partially destroyed.
  • if the apparatus itself had a likelihood of affecting the future state of the particle, then, in a very real sense, a measurement will be made, and the interference more or less destroyed.

We would like to define a measure of the degree of entanglement of a given state. For example, a pair of electrons could be completely entangled whilst in the singlet state, say, and completely non-entangled whilst in the $\ket{uu}$ state.

The measure we shall use (there are others) is called entanglement entropy. Entropy, in general, is not just a property of the system alone, but also reflects the knowledge about the particular state the system is in. In particular, the less you know about a system, the more entropy it has and conversely, the more you know about a system, the less entropy it has.